(Quantum) Fractional Brownian Motion and Multifractal Processes under the Loop of a Tensor Networks
Abstract
We derive fractional Brownian motion and stochastic processes with multifractal properties using a framework of network of Gaussian conditional probabilities. This leads to the derivation of new representations of fractional Brownian motion. These constructions are inspired from renormalization. The main result of this paper consists of constructing each increment of the process from twodimensional gaussian noise inside the lightcone of each seperate increment. Not only does this allows us to derive fractional Brownian motion, we can introduce extensions with multifractal flavour. In another part of this paper, we discuss the use of the multiscale entanglement renormalization ansatz (MERA), introduced in the study critical systems in quantum spin lattices, as a method for sampling integrals with respect to such multifractal processes. After proper calibration, a MERA promises the generation of a sample of size $N$ of a multifractal process in the order of $O(N\log(N))$, an improvement over the known methods, such as the Cholesky decomposition and the circulant methods, which scale between $O(N^2)$ and $O(N^3)$.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 DOI:
 10.48550/arXiv.1602.00924
 arXiv:
 arXiv:1602.00924
 Bibcode:
 2016arXiv160200924D
 Keywords:

 Quantum Physics;
 Physics  Computational Physics;
 Statistics  Applications
 EPrint:
 10 pages